## Event Analysis

### Introduction

The routines in this section are intended to be used to analyze event properties. As such they are not part of the main event generation chain, but can be used in comparisons between Monte Carlo events and real data. They are rather free-standing, but assume that input is provided in the PYTHIA 8 `Event` format, and use a few basic facilities such as four-vectors. Their ordering is mainly by history; for current LHC applications the final one, `SlowJet`, is of special interest.

In addition to the methods presented here, there is also the possibility to make use of external jet finders .

### Sphericity

The standard sphericity tensor is S^{ab} = (sum_i p_i^a p_i^b) / (sum_i p_i^2) where the sum i runs over the particles in the event, a, b = x, y, z, and p without such an index is the absolute size of the three-momentum . This tensor can be diagonalized to find eigenvalues and eigenvectors.

The above tensor can be generalized by introducing a power r, such that S^{ab} = (sum_i p_i^a p_i^b p_i^{r-2}) / (sum_i p_i^r) In particular, r = 1 gives a linear dependence on momenta and thus a collinear safe definition, unlike sphericity.

To do sphericity analyses you have to set up a `Sphericity` instance, and then feed in events to it, one at a time. The results for the latest event are available as output from a few methods.

create a sphericity analysis object, where is the power r defined above, i.e. gives Sphericity, and gives the linear form. tells which particles are analyzed, all final-state particles, all observable final-state particles, i.e. excluding neutrinos and other particles without strong or electromagnetic interactions (the `isVisible()` particle method), and only charged final-state particles.
perform a sphericity analysis, where is an object of the `Event` class, most likely the `pythia.event` one.
If the routine returns `false` the analysis failed, e.g. if too few particles are present to analyze.

After the analysis has been performed, a few methods are available to return the result of the analysis of the latest event:

gives the sphericity (or equivalent if r is not 2),
gives the aplanarity (with the same comment),
gives one of the three eigenvalues for i = 1, 2 or 3, in descending order,
gives the matching normalized eigenvector, as a `Vec4` with vanishing time/energy component.
provides a listing of the above information.

There is also one method that returns information accumulated for all the events analyzed so far.

tells the number of times `analyze(...)` failed to analyze events, i.e. returned `false`.

### Thrust

Thrust is obtained by varying the thrust axis so that the longitudinal momentum component projected onto it is maximized, and thrust itself is then defined as the sum of absolute longitudinal momenta divided by the sum of absolute momenta. The major axis is found correspondingly in the plane transverse to thrust, and the minor one is then defined to be transverse to both. Oblateness is the difference between the major and the minor values.

The calculation of thrust is more computer-time-intensive than e.g. linear sphericity, introduced above, and has no specific advantages except historical precedent. In the PYTHIA 6 implementation the search was sped up at the price of then not being guaranteed to hit the absolute maximum. The current implementation studies all possibilities, but at the price of being slower, with time consumption for an event with n particles growing like n^3.

To do thrust analyses you have to set up a `Thrust` instance, and then feed in events to it, one at a time. The results for the latest event are available as output from a few methods.

create a thrust analysis object, where tells which particles are analyzed, all final-state particles, all observable final-state particles, i.e. excluding neutrinos and other particles without strong or electromagnetic interactions (the `isVisible()` particle method), and only charged final-state particles.
perform a thrust analysis, where is an object of the `Event` class, most likely the `pythia.event` one.
If the routine returns `false` the analysis failed, e.g. if too few particles are present to analyze.

After the analysis has been performed, a few methods are available to return the result of the analysis of the latest event:

gives the thrust, major, minor and oblateness values, respectively,
gives the matching normalized event-axis vectors, for i = 1, 2 or 3 corresponding to thrust, major or minor, as a `Vec4` with vanishing time/energy component.
provides a listing of the above information.

There is also one method that returns information accumulated for all the events analyzed so far.

tells the number of times `analyze(...)` failed to analyze events, i.e. returned `false`.

### ClusterJet

ClusterJet
(a.k.a. `LUCLUS` and `PYCLUS`) is a clustering algorithm of the type used for analyses of e^+e^- events, see the PYTHIA 6 manual. All visible particles in the events are clustered into jets. A few options are available for some well-known distance measures. Cutoff distances can either be given in terms of a scaled quadratic quantity like y = pT^2/E^2 or an unscaled linear one like pT.

Note that we have deliberately chosen not to include the e^+e^- equivalents of the Cambridge/Aachen and anti-kRT algorithms. These tend to be good at clustering the densely populated (in angle) cores of jets, but less successful for the sparsely populated transverse regions, where many jets may come to consist of a single low-momentum particle. In hadron collisions such jets could easily be disregarded, while in e^+e^- annihilation all particles derive back from the hard process.

To do jet finding analyses you have to set up a `ClusterJet` instance, and then feed in events to it, one at a time. The results for the latest event are available as output from a few methods.

create a `ClusterJet` instance, where distance measure, to be provided as a character string (actually, only the first character is necessary) the Lund pT distance, the JADE mass distance, and the Durham kT measure. tells which particles are analyzed, all final-state particles, all observable final-state particles, i.e. excluding neutrinos and other particles without strong or electromagnetic interactions (the `isVisible()` particle method), and only charged final-state particles. masses assumed for the particles used in the analysis all massless, photons are massless while all others are assigned the pi+- mass, and all given their correct masses. perform or not a early preclustering step, where nearby particles are lumped together so as to speed up the subsequent normal clustering. reassign all particles to the nearest jet each time after two jets have been joined.
performs a jet finding analysis, where is an object of the `Event` class, most likely the `pythia.event` one. is the cutoff joining scale, below which jets are joined. Is given in quadratic dimensionless quantities. Either `yScale` or `pTscale` must be set nonvanishing, and the larger of the two dictates the actual value. is the cutoff joining scale, below which jets are joined. Is given in linear quantities, such as pT or m depending on the measure used, but always in units of GeV. Either `yScale` or `pTscale` must be set nonvanishing, and the larger of the two dictates the actual value. the minimum number of jets to be reconstructed. If used, it can override the `yScale` and `pTscale` values. the maximum number of jets to be reconstructed. Is not used if below `nJetMin`. If used, it can override the `yScale` and `pTscale` values. Thus e.g. `nJetMin = nJetMax = 3` can be used to reconstruct exactly 3 jets.
If the routine returns `false` the analysis failed, e.g. because the number of particles was smaller than the minimum number of jets requested.

After the analysis has been performed, a few `ClusterJet` class methods are available to return the result of the analysis:

gives the number of jets found, with jets numbered 0 through `size() - 1`.
gives a `Vec4` corresponding to the four-momentum defined by the sum of all the contributing particles to the i'th jet.
the number of particles that have been clustered into the i'th jet.
gives the index of the jet that the particle i of the event record belongs to,
provides a listing of the reconstructed jets.
the number of most recent clustering scales that have been stored for readout with the next method. Normally this would be five, but less if fewer clustering steps occurred.
clustering scales, with `distance(0)` being the most recent one, i.e. normally the highest, up to `distance(4)` being the fifth most recent. That is, with n being the final number of jets, `ClusterJet::size()`, the scales at which n+1 jets become n, n+2 become n+1, and so on till n+5 become n+4. Nonexisting clustering scales are returned as zero. The physical interpretation of a scale is as provided by the respective distance measure (Lund, JADE, Durham).

There is also one method that returns information accumulated for all the events analyzed so far.

tells the number of times `analyze(...)` failed to analyze events, i.e. returned `false`.

### CellJet

CellJet
(a.k.a. `PYCELL`) is a simple cone jet finder in the UA1 spirit, see the PYTHIA 6 manual. It works in an (eta, phi, eT) space, where eta is pseudorapidity, phi azimuthal angle and eT transverse energy. It will draw cones in R = sqrt(Delta-eta^2 + Delta-phi^2) around seed cells. If the total eT inside the cone exceeds the threshold, a jet is formed, and the cells are removed from further analysis. There are no split or merge procedures, so later-found jets may be missing some of the edge regions already used up by previous ones. Not all particles in the event are assigned to jets; leftovers may be viewed as belonging to beam remnants or the underlying event. It is not used by any experimental collaboration, but is closely related to the more recent and better theoretically motivated anti-kT algorithm Cac08.

To do jet finding analyses you have to set up a `CellJet` instance, and then feed in events to it, one at a time. Especially note that, if you want to use the options where energies are smeared in order so emulate detector imperfections, you must hand in an external random number generator, preferably the one residing in the `Pythia` class. The results for the latest event are available as output from a few methods.

create a `CellJet` instance, where the maximum +-pseudorapidity that the detector is assumed to cover. the number of equal-sized bins that the +-etaMax range is assumed to be divided into. the number of equal-sized bins that the phi range +-pi is assumed to be divided into. tells which particles are analyzed, all final-state particles, all observable final-state particles, i.e. excluding neutrinos and other particles without strong or electromagnetic interactions (the `isVisible()` particle method), and only charged final-state particles. strategy to smear the actual eT bin by bin, no smearing, smear the eT according to a Gaussian with width resolution * sqrt(eT), with the Gaussian truncated at 0 and upperCut * eT, smear the e = eT * cosh(eta) according to a Gaussian with width resolution * sqrt(e), with the Gaussian truncated at 0 and upperCut * e. see above. see above. completely neglect all bins with an eT < threshold. the random-number generator used to select the smearing described above. Must be handed in for smearing to be possible. If your `Pythia` class instance is named `pythia`, then `&pythia.rndm` would be the logical choice.
performs a jet finding analysis, where is an object of the `Event` class, most likely the `pythia.event` one. is the minimum transverse energy inside a cone for this to be accepted as a jet. is the size of the cone in (eta, phi) space drawn around the geometric center of the jet. the minimum eT in a cell for this to be acceptable as the trial center of a jet.
If the routine returns `false` the analysis failed, but currently this is not foreseen ever to happen.

After the analysis has been performed, a few `CellJet` class methods are available to return the result of the analysis:

gives the number of jets found, with jets numbered 0 through `size() - 1`,
gives the eT of the i'th jet, where jets have been ordered with decreasing eT values,
gives the eta and phi coordinates of the geometrical center of the i'th jet,
gives the eta and phi coordinates of the eT-weighted center of the i'th jet,
gives the number of particles clustered into the i'th jet,
gives a `Vec4` corresponding to the four-momentum defined by the eT and the weighted center of the i'th jet,
gives a `Vec4` corresponding to the four-momentum defined by the sum of all the contributing cells to the i'th jet, where each cell contributes a four-momentum as if all the eT is deposited in the center of the cell,
gives the invariant mass of the i'th jet, defined by the `pMassive` above,
provides a listing of the above information (except `pMassless`, for reasons of space).

There is also one method that returns information accumulated for all the events analyzed so far. tells the number of times `analyze(...)` failed to analyze events, i.e. returned `false`.

### SlowJet

SlowJet
is a simple program for doing jet finding according to either of the kT, anti-kT, and Cambridge/Aachen algorithms, in a cylindrical coordinate frame. The name is obviously an homage to the `FastJet` program Cac06, Cac12. That package contains many more algorithms, with many more options, and, above all, is much faster. Therefore `SlowJet` is not so much intended for massive processing of data or Monte Carlo files as for simple first tests. Nevertheless, within the advertised capabilities of `SlowJet`, it has been checked to find identically the same jets as `FastJet`. The time consumption typically is around or below that to generate an LHC pp event in the first place, so is not prohibitive. But the time rises rapidly for large multiplicities, so obviously `SlowJet` can not be used for tricks like distributing a dense grid of pseudoparticles to be able to define jet areas, like `FastJet` can, and also not for events with much pileup or other noise.

The recent introduction of `fjcore`, containing the core functionality of `FastJet` in a very much smaller package, has changed the conditions. It now is possible (even encouraged by the authors) to distribute the two `fjcore` files as part of the PYTHIA package. Therefore the `SlowJet` class doubles as a convenient front end to `fjcore`, managing the conversion back and forth between PYTHIA and `FastJet` variables. Some applications may still benefit from using the native codes, but the default now is to let `SlowJet` call on `fjcore` for the jet finding.

The first step is to decide which particles should be included in the analysis, and with what four-momenta. The `SlowJet` constructor allows to pick a maximum pseudorapidity defined by the extent of the assumed detector, to pick among some standard options of which particles to analyze, and to allow for some standard mass assumptions, like that all charged particles have the pion mass. Obviously this is only a restricted set of possibilities.

Full flexibility can be obtained by deriving from the base class `SlowJetHook` to write your own `include` method. This will be presented with one final-state particle at a time, and should return `true` for those particles that should be analyzed. It is also possible to return modified four-momenta and masses, to take into account detector smearing effects or particle identity misassignments, but you must respect E^2 - p^2 = m^2.

Alternatively you can modify the event record itself, or a copy of it (if you want to keep the original intact). For instance, only final particles are considered in the analysis, i.e. particles with positive status code, so negative status code should then be assigned to those particles that you do not want to see analyzed. Again four-momenta and masses can be modified, subject to E^2 - p^2 = m^2.

The jet reconstructions is then based on sequential recombination with progressive removal, using the E recombination scheme. To be more specific, the algorithm works as follows.

1. Each particle to be analyzed defines an original cluster. It has a well-defined four-momentum and mass at input. From this information the triplet (pT, y, phi) is calculated, i.e. the transverse momentum, rapidity and azimuthal angle of the cluster.
2. Define distance measures of all clusters i to the beam d_iB = pT_i^2p and of all pairs (i,j) relative to each other d_ij = min( pT_i^2p, pT_j^2p) DeltaR_ij^2 / R^2 where DeltaR_ij^2 = (y_i - y_j)^2 + (phi_i - phi_j)^2. The jet algorithm is determined by the user-selected p value, where p = -1 corresponds to the anti-kT one, p = 0 to the Cambridge/Aachen one and p = 1 to the kT one. Also R is chosen by the user, to give an approximate measure of the size of jets. However, note that jets need not have a circular shape in (y, phi) space, so R can not straight off be interpreted as a jet radius.
3. Find the smallest of all d_iB and d_ij.
4. If this is a d_iB then cluster i is removed from the clusters list and instead added to the jets list. Optionally, a pTjetMin requirement is imposed, where only clusters with pT > pTjetMin are added to the jets list. If so, some of the analyzed particles will not be part of any final jet.
5. If instead the smallest measure is a d_ij then the four-momenta of the i and j clusters are added to define a single new cluster. Convert this four-momentum to a new (pT, y, phi) triplet and update the list of d_iB and d_ij.
6. Return to step 3 until no clusters remain.

To do jet finding analyses you first have to set up a `SlowJet` instance, where the arguments of the constructor specifies the details of the subsequent analyses. Thereafter you can feed in events to it, one at a time, and have them analyzed by the `analyze` method. Information on the resulting jets can be extracted by a few different methods. The minimal procedure only requires one call per event to do the analysis. We will begin by presenting it, and only afterwards some extensions.

create a `SlowJet` instance, where tells (half) the power of the transverse-momentum dependence of the distance measure, the anti-kT algorithm, the Cambridge/Aachen algorithm, and the kT algorithm. the R size parameter, which is crudely related to the radius of the jet cone in (y, phi) space around the center of the jet. the minimum transverse momentum required for a cluster to become a jet. By default all clusters become jets, and therefore all analyzed particles are assigned to a jet. For comparisons with perturbative QCD, however, it is only meaningful to consider jets with a significant pT. the maximum +-pseudorapidity that the detector is assumed to cover. If you pick a value above 20 there is assumed to be full coverage (obviously only meaningful for theoretical studies). tells which particles are analyzed, all final-state particles, all observable final-state particles, i.e. excluding neutrinos and other particles without strong or electromagnetic interactions (the `isVisible()` particle method), and only charged final-state particles. masses assumed for the particles used in the analysis all massless, photons are massless while all others are assigned the pi+- mass, and all given their correct masses. gives the possibility to send in your own selection routine for which particles should be part of the analysis; see further below on the `SlowJetHook` class. If this pointer is sent in nonzero, `etaMax` and `massSet` are disregarded, and `select` only gives the basic selection, to which the user can add further requirements. choice of code used for finding the jets. Does not affect the outcome of the analysis, but only the speed, and some more specialized options. use the `fjcore` package of `FastJet 3.0.5`. use the native `SlowJet` implementation, which gives a slower jet finding, but allows some extra options of step-by-step jet joining. definition of R distance between two jets. This switch is only meaningful for `useFJcore = false`; within the `fjcore` package the standard option below is always used. standard, as described above, DeltaR_ij^2 = (y_i - y_j)^2 + (phi_i - phi_j)^2. alternative, DeltaR_ij^2 = 2 (cosh(y_i - y_j) - cos(phi_i - phi_j)), which corresponds to the rim of the "deformed cone" giving a constant invariant mass between the two partons considered (for fixed masses and transverse momenta).
performs a jet finding analysis, where is an object of the `Event` class, most likely the `pythia.event` one.
If the routine returns `false` the analysis failed, but currently this is not foreseen ever to happen.

After the analysis has been performed, a few `SlowJet` class methods are available to return the result of the analysis:

gives the original number of particles (and thus clusters) that the analysis starts with.
gives the number of jets found, with jets numbered 0 through `sizeJet() - 1`, and ordered in terms of decreasing transverse momentum values w.r.t. the beam axis,
gives the transverse momentum pT of the i'th jet,
gives the rapidity y and azimuthal angle phi of the center of the i'th jet (defined by the vector sum of constituent four-momenta),
gives a `Vec4` corresponding to the four-momentum sum of the particles assigned to the i'th jet, and the invariant mass of this four-vector,
gives the number of particles clustered into the i'th jet,
gives a list of the indices of the particles that have been clustered into the i'th jet,
gives a list of the indices of the particles that have been clustered into the i'th cluster, at the current stage of the clustering process,
gives the index of the jet that the particle i of the event record belongs to, or -1 if there is no jet containing particle i,
removes the i'th jet,
provides a listing of the basic jet information from above.

These are the basic methods. For more sophisticated usage it is possible to trace the clustering, one step at a time. It requires the native jet finding code, `useFJcore = false` in the constructor. If so, the `setup` method should be used to read in the event and find the initial smallest distance. Each subsequent `doStep` will then do one cluster joining and find the new smallest distance. You can therefore interrogate which clusters will be joined next before the joining actually is performed. Alternatively you can take several steps in one go, or take steps down to a predetermined number of jets plus remaining clusters.

selects the particles to be analyzed, calculates initial distances, and finds the initial smallest distance. is an object of the `Event` class, most likely the `pythia.event` one.
If the routine returns `false` the setup failed, but currently this is not foreseen ever to happen.
do the next step of the clustering. This can either be that two clusters are joined to one, or that a cluster is promoted to a jet (which is discarded if its pT value is below `pTjetMin`).
The routine will only return `false` if there are no clusters left, or if `useFJcore = true`.
calls the `doStep()` method `nStep` times, if possible. Will return `false` if the list of clusters is emptied before then. The stored jet information is still perfectly fine; it is only the number of steps that is wrong. Will also return `false` if `useFJcore = true`.
calls the `doStep()` method until a total of `nStop` jet and cluster objects remain. Will return `false` if this is not possible, specifically if the number of objects already is smaller than `nStop` when the method is called. The stored jet and cluster information is still perfectly fine; it only does not have the expected multiplicity. Will also return `false` if `useFJcore = true`.
gives the total current number of jets and clusters. The jets are numbered 0 through `sizeJet() - 1`, while the clusters are numbered `sizeJet()` through `sizeAll() - 1`. (Internally jets and clusters are represented by two separate arrays, but are here presented in one flat range.) Note that the jets are ordered in decreasing pT, while the clusters are not ordered. When `useFJcore = true` there are no intermediate steps, and thus the number of clusters is zero (after jet finding).

With this extension, the methods `double pT(int i)`, `double y(int i)`, `double phi(int i)`, `Vec4 p(int i)`, `double m(int i)` and `int multiplicity(int i)` can be used as before. Furthermore, `list()` generalizes

provides a listing of the above information. lists both jets and clusters if `true`, else only jets.

Three further methods can be used to check what will happen next.

if the next step is to join two clusters, then the methods give the (i,j, d_ij) values, if instead to promote a cluster to a jet then (i, -1, d_iB). If no clusters remain then (-1, -1, 0.). Note that the cluster numbers are offset as described above, i.e. they begin at `sizeJet()`, which of course easily could be subtracted off. Also note that the jet and cluster lists are (moderately) reshuffled in each new step. When `useFJcore = true` there are no intermediate steps, and thus these methods do not return meaningul information.

Finally, and separately, the `SlowJetHook` class can be used for a more smart selection of which particles to include in the analysis. For instance, isolated and/or high-pT muons, electrons and photons should presumably be identified separately at an early stage, and then not clustered to jets.

Technically, it works like with User Hooks. That is, PYTHIA contains the base class. You write a derived class. In the main program you create an instance of this class, and hand it in to `SlowJet`; in this case already as part of the constructor.

The following methods should be defined in your derived class.

the constructor and destructor need not do anything, and if so you need not write your own destructor.
is the main method that you will need to write. It will be called once for each final-state particle in an event, subject to the value of the `select` switch in the `SlowJet` constructor. The value `select = 2` may be convenient since then you do not need to remove e.g. neutrinos yourself, but use `select = 1` for full control. The method should then return `true` if you want to see particle included in the jet clustering, and `false` if not. is the index in the event record of the currently studied particle. is an object of the `Event` class, most likely the `pythia.event` one, where the currently studied particle is found. is at input the four-momentum of the currently studied particle. You can change the values, e.g. to take into account energy smearing in the detector, to define the initial cluster value, without corrupting the event record itself. is at input the mass of the currently studied particle. You can change the value, e.g. to take into account particle misidentification, to define the initial cluster value, without corrupting the event record itself. Note that the changes of `pSel` and `mSel` must be coordinated such that E^2 - p^2 = m^2 holds.

It is also possible to define further methods of your own. One such could e.g. be called directly in the main program before the `analyze` method is called, to identify and bookkeep some event properties you may not want to reanalyze for each individual particle.